#7763425

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7483 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14,19,16)(15,17,21,23)(18,20,26,24)(22,25,27,28);(1,2)(3,4)(13,15,19,21)(14,17,16,23)(18,22,26,27)(20,25,24,28);(6,9)(10,11)(14,16)(15,21)(18,26)(25,28);(5,6)(7,10)(8,9)(11,12)(13,16,19,14)(15,17,21,23)(18,24,26,20)(22,25,27,28);(5,7,8,12)(6,10,9,11)(13,17)(14,15)(16,21)(18,25)(19,23)(20,22)(24,27)(26,28);(13,18,19,26)(14,20,16,24)(15,22,21,27)(17,25,23,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=i,e^2=hi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236858, #72238154, #72238245
maximal normal subgroup of	#72236858, #72238154, #72238245

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7670621);